SOCIAL CHOICE WITH FUZZY PREFERENCES RICHARD BARRETT AND MAURICE SALLES 1. introduction The most important concept of social choice theory is probably the concept of prefer- ence, be it individual preference or social preference. Typically, preferences are crisp, i.e., given by a binary relation over the set of options (social states, candidates, etc.). Then, for two options, either one is preferred to the other, or there is an indifference between them or there is no relation between them. In most cases, the possibility that there is no relation between them is excluded. The binary relation is then complete. Several au- thors, including Sen (1992), have considered that incompleteness was a way to deal with ambiguity. However, when the options are uncertain we must adopt preferences related in some way to probabilities and when they are complex, for instance when each option has manifold characteristics, we must try to take account of the vagueness this entails. To deal with vagueness, various authors have had recourse to, roughly speaking, three different techniques. The first is probability theory. Fishburn (1998) writes: ‘Vague preferences and wavering judgments about better, best, or merely satisfactory alternatives lead naturally to theories based on probabilistic preference and probabilistic choice.’ The second is fuzzy set theory. Each of these two theories has champions that apparently strongly disagree. There are, however, several signs indicating that the scientific war should eventually end (see Ross, Booker and Parkinson (2002) and the forewords in this book by Zadeh and by Suppes). The third is due to philosophers. Vagueness is an important topic within analytic philosophy (Keefe and Smith (1996), Burns (1991), Williamson (1994), Keefe (2000)). Philosophers have mainly been concerned with predicates such as tall, small, red, bald, heap, and solutions to the sorites paradox. (For instance, a nice example for the predicate ‘small’ is Wang’s paradox discussed by Dummett (1975). Consider the inductive argument: 0 is small; If n is small, n + 1 is small: Therefore every number is small.) Although there is little agreement among them about the nature of vagueness, a theory has been designed to deal with it: supervaluation theory. In an interesting paper, Broome (1997) has jointly considered vagueness and incompleteness. As social choice theorists, we are perhaps more interested in vagueness of relations than the kind of predicates mentioned above and most of the papers dealing with vagueness in choice theory have used fuzzy sets. The purpose of this chapter is to present some of the main results obtained on the aggregation of fuzzy preferences. Regarding the possible interpretations of preferences, it seems to us that we must confine ourselves to what Sen (1983, 2002) calls the outcome evaluation. Preference is about the fact that an option is judged to be a better state of affairs than another option. In Sen’s typology, the two other interpretations are about choices, normative or descriptive. In particular, our analysis is not well adapted to voting, since the voters’ ballot papers are not vague and the results of the election are not vague either, even if, before designing a crisp preference, each voter had rather fuzzy preferences. Date: August 10, 2004. We are most grateful to Prasanta Pattanaik for our joint work on this topic over the years. Thanks also to a referee for useful suggestions and to Dinko Dimitrov for his careful reading and for providing corrections. 1 2 RICHARD BARRETT AND MAURICE SALLES After introducing the concepts of fuzzy preference in Section 2, Section 3 will be de- voted to Arrovian aggregation problems and Section 4 to other aspects, including a fuzzy treatment of Sen’s impossibility of a Paretian liberal and the first results about fuzzy aggregation in economic environments. 2. fuzzy preferences In (naive) set theory, given a set X, a subset A and an element x ∈ X, either x ∈ A or x∈ / A. Belonging to the subset can be defined by a function b from X to {0, 1}, where x ∈ A is equivalent to b(x) = 1 and x∈ / A is equivalent to b(x) = 0. If the subset A refers to the description of some semantic concepts (events or phenomena or statements), there might be no clear-cut way to assert that an element is or is not in this subset. Classical examples are the set of tall men, the set of intelligent women or the set of beautiful spiders. The basic idea of replacing {0, 1} by [0, 1] as the set where the membership function takes its values is due to Zadeh. However, the origin of this is probably older, taking us back at least toLukasiewicz who introduced many-valued logic in 1920. For excellent mathemati- cal introductions, we recommend Dubois and Prade (1980) and Nguyen and Walker (2000). For a fuzzy binary relation, the membership function associates a number in [0, 1] to an ordered pair of options (x, y). The interpretation of a number α ∈ [0, 1] associated to (x, y) can be the degree of intensity with which x is preferred to y is α or the degree of truth that x is preferred to y is α. Though in some cases the first interpretation is possible and amounts to considering strength of preference, the second interpretation is always possible and is compulsory if, at some stage, fuzzy connectives (and, or ...) have been introduced. The fact that some positive value α is associated to (x, y) does not entail, in general, that the value 0 must be assigned to (y, x). For instance, if we consider two versions of Beethoven’s Grosse Fuge, say, by the Juilliard Quartet and the Berg Quartet, we may have mixed and conflicting feelings. We may prefer to some extent the Berg version because of its energy but have also some preference for the Juilliard because of its poetry. This will entail that some value α ∈]0, 1] be given to the preference for the Berg version over the Juilliard, but also that some value β ∈]0, 1] be given to the preference for the Juilliard version over the Berg. Moreover, on the basis of this example, it might seem strange to describe fuzziness by assigning a precise number. The logician Alasdair Urquhart (2001) has rightly observed : One immediate objection that presents itself to this line of approach is the extremely artificial nature of the attaching of precise numerical values to sentences like ‘73 is a large number’ or ‘Picasso’s Guernica is beautiful’. In fact, it seems plausible to say that the nature of vague pred- icates precludes attaching precise numerical values just as much as it precludes attaching precise classical truth values. One way to avoid this difficulty is to replace [0, 1] ordered by ≥ by some set of qual- itative elements ordered by a complete preorder. For instance, to give some intuition in the context of preference, the elements could be interpreted as being ‘not at all’, ‘insignif- icantly’, ‘a little’, ‘mildly’, ‘much’, ‘very much’, ‘definitely’, etc. We will consider both cases in the following subsections. 2.1. Fuzzy preferences: numerical values. We will stick in this chapter to the notions and properties used in the fuzzy aggregation literature. There are many notions about transitivity and choice which will not be introduced (see, for instance, Barrett, Pattanaik and Salles (1990), Basu, Deb and Pattanaik (1992), Dasgupta and Deb ((1991), (1996), (2001)), Salles (1998)). SOCIAL CHOICE WITH FUZZY PREFERENCES 3 Let X be the set of alternatives with #X ≥ 3. We will consider two types of fuzzy binary relations. Strict binary relations as described here were introduced in Barrett, Pattanaik and Salles (1986), hereafter denoted by BPS, and weak binary relations were introduced by Dutta (1987). Definition 1. A fuzzy binary relation over X is a function h : X × X → [0, 1]. Definition 2. A fuzzy binary relation p is a BPS-fuzzy strict preference if for all distinct x, y, z ∈ X, (i) p(x, x) = 0 (ii) p(x, y) = 1 ⇒ p(y, x) = 0 (iii) p(x, y) > 0 and p(y, z) > 0 ⇒ p(x, z) > 0. Definition 3. A BP S-fuzzy strict preference p is BPS-complete if for all distinct x, y ∈ X, p(x, y) > 0 or p(y, x) > 0. p(x, y) can be interpreted as the degree of intensity with which x is preferred to y (or, see above, the degree of truth that x is preferred to y). (i) expresses that we con- sider strict preferences. (ii) means that when strict preferences are definite (in some sense non-fuzzy), they are asymmetric. (iii) is a rather mild transitivity property. It is now well known that many transitivity concepts are available for fuzzy binary relations (Das- gupta and Deb, 1996). In particular, the most widely used concept, max-min transitivity, which states that given any x, y, z ∈ X, p(x, z) ≥ min(p(x, y), p(y, z)) is stronger than (iii). Subramanian (1987) uses two different fuzzy strict preferences one of which is a variant of BP S-fuzzy strict preferences. S1 Definition 4. A fuzzy strict preference p is a S1-fuzzy strict preference if it is a BP S-fuzzy strict preference for which for all distinct x, y, z ∈ X, pS1 (x, y) = pS1 (y, x) = pS1 (y, z) = pS1 (z, y) = 0 ⇒ pS1 (x, z) = pS1 (z, x) = 0. S2 A fuzzy strict preference p is a S2-fuzzy strict preference if it satisfies properties (i) and (ii) of Definition 2, and if S2 S2 S2 S2 for all distinct x1, x2, ..., xk ∈ X, p (x1, x2) > p (x2, x1) and p (x2, x3) > p (x3, x2) S2 S2 S2 S2 and ..